Octal Hexadecimal Number Systems Digital Electronics What More

Octal & Hexadecimal Number Systems Digital Electronics

What, More Number Systems? Why do we need more number systems? Humans understand decimal. Digital electronics (computers) understand binary. Since computers have 32, 64, and even 128 bit busses, displaying numbers in binary is cumbersome. 2

What, More Number Systems? Data on a 32 bit data bus would look like the following: 0110 1001 0111 0001 0011 0100 1010 Hexadecimal (base 16) and octal (base 8) number systems are used to represent binary data in a more compact form. This presentation will present an overview of the process for converting numbers between the decimal number system and the hexadecimal & octal number systems. 3

What, More Number Systems? This presentation will present an overview of the process for converting numbers between the decimal number system and the hexadecimal & octal number systems. 4

Converting To and From Decimal 10 Successive Division Octal 8 0123456789 Weighted Multiplication Successive Division Successive Weighted Division Multiplication Weighted Multiplication Hexadecimal 16 0123456789 ABCDEF 01234567 Binary 2 01 5

Counting. . . 2, 8, 10, 16 Decimal Binary Octal Hexadecimal 0 00000 0 0 1 00001 1 1 2 00010 2 2 3 00011 3 3 4 00100 4 4 5 00101 5 5 6 00110 6 6 7 00111 7 7 8 01000 10 8 9 01001 11 9 10 01010 12 A 11 01011 13 B 12 01100 14 C 13 01101 15 D 14 01110 16 E 15 01111 17 F 16 10000 20 10 17 10001 21 11 18 10010 22 12 19 10011 23 13 6

Review: Decimal ↔ Binary Base 10 DECIMAL Successive Division Base 2 BINARY a) Divide the decimal number by 2; the remainder is the LSB of the binary number. b) If the quotient is zero, the conversion is complete. Otherwise repeat step (a) using the quotient as the decimal number. The new remainder is the next most significant bit of the binary number. 7

Review: Decimal ↔ Binary Base 2 BINARY Weighted Multiplication Base 10 DECIMAL a) Multiply each bit of the binary number by its corresponding bit-weighting factor (i. e. , Bit 0→ 20=1; Bit-1→ 21=2; Bit-2→ 22=4; etc). b) Add up all of the products in step (a) to get the decimal number. 8

Conversion Process Decimal ↔ Base. N (Any base including Binary 2, Octal 8, Hexidecimal 16) Base 10 DECIMAL Successive Division Base. N ANY BASE a) Divide the decimal number by N; the remainder is the LSB of the ANY BASE Number. b) If the quotient is zero, the conversion is complete. Otherwise repeat step (a) using the quotient as the decimal number. The new remainder is the next most significant bit of the ANY BASE number. 9

Conversion Process Decimal ↔ Base. N (Any base including Binary 2, Octal 8, Hexidecimal 16) Base. N ANY BASE Weighted Multiplication Base 10 DECIMAL a) Multiply each bit of the ANY BASE number by its corresponding bit-weighting factor (i. e. , Bit 0→N 0; Bit-1→N 1; Bit-2→N 2; etc). b) Add up all of the products in step (a) to get the decimal number. 10

Decimal ↔ Octal Conversion The Process: Successive Division Divide the decimal number by 8; the remainder is the LSB of the octal number. If the quotation is zero, the conversion is complete. Otherwise repeat step (a) using the quotation as the decimal number. The new remainder is the next most significant bit of the octal number. 11

Decimal ↔ Octal Conversion Example: Convert the decimal number 9410 into its octal equivalent. 12

Decimal ↔ Octal Conversion Example: Convert the decimal number 9410 into its octal equivalent. 13

Decimal ↔ Octal Conversion Example: Convert the decimal number 9410 into its octal equivalent. 14

Decimal ↔ Octal Conversion Example: Convert the decimal number 9410 into its octal equivalent. 9410 = 1368 15

Example: Decimal → Octal Example: Convert the decimal number 18910 into its octal equivalent. 16

Example: Decimal → Octal Example: Convert the decimal number 18910 into its octal equivalent. 17

Example: Decimal → Octal Example: Convert the decimal number 18910 into its octal equivalent. 18

Example: Decimal → Octal Example: Convert the decimal number 18910 into its octal equivalent. 19

Example: Decimal → Octal Example: Convert the decimal number 18910 into its octal equivalent. 18910 = 2758 20

Octal ↔ Decimal Process The Process: Weighted Multiplication Multiply each bit of the Octal Number by its corresponding bit-weighting factor (i. e. , Bit 0→ 80=1; Bit-1→ 81=8; Bit-2→ 82=64; etc. ). Add up all of the products in step (a) to get the decimal number. 21

Octal ↔ Decimal Process Example: Convert the octal number 1368 into its decimal equivalent. 82 81 80 22

Octal ↔ Decimal Process Example: Convert the octal number 1368 into its decimal equivalent. 82 81 80 64 8 1 23

Octal ↔ Decimal Process Example: Convert the octal number 1368 into its decimal equivalent. 82 81 80 64 8 1 Bit-Weighting Factors 24

Octal ↔ Decimal Process Example: Convert the octal number 1368 into its decimal equivalent. 1 3 6 82 81 80 64 8 1 Bit-Weighting Factors 25

Octal ↔ Decimal Process Example: Convert the octal number 1368 into its decimal equivalent. 1 3 6 82 81 80 64 8 1 Bit-Weighting Factors 6 26

Octal ↔ Decimal Process Example: Convert the octal number 1368 into its decimal equivalent. 1 3 6 82 81 80 64 8 1 24 6 Bit-Weighting Factors 27

Octal ↔ Decimal Process Example: Convert the octal number 1368 into its decimal equivalent. 1 3 6 82 81 80 64 8 1 64 24 6 Bit-Weighting Factors 28

Octal ↔ Decimal Process Example: Convert the octal number 1368 into its decimal equivalent. 1 3 6 82 81 80 64 8 1 64 + 24 + 6 Bit-Weighting Factors = 29

Octal ↔ Decimal Process Example: Convert the octal number 1368 into its decimal equivalent. 1 3 6 82 81 80 64 8 1 64 + 24 + 6 Bit-Weighting Factors = 9410 30

Decimal ↔ Hexadecimal Conversion The Process: Successive Division Divide the decimal number by 16; the remainder is the LSB of the hexadecimal number. If the quotient is zero, the conversion is complete. Otherwise repeat step (a) using the quotation as the decimal number. The new remainder is the next most significant bit of the hexadecimal number. 31

Decimal ↔ Hexadecimal Conversion Counting in Hexadecimal 0 1 2 3 4 5 6 7 8 9 A B C D E F where… A = 10; B = 11; C = 12 D = 13; E = 14; F = 15 32

Example: Decimal → Hexadecimal Example: Convert the decimal number 42910 into its hexadecimal equivalent. 33

Example: Decimal → Hexadecimal Example: Convert the decimal number 42910 into its hexadecimal equivalent. 34

Example: Decimal → Hexadecimal Example: Convert the decimal number 42910 into its hexadecimal equivalent. 35

Example: Decimal → Hexadecimal Example: Convert the decimal number 42910 into its hexadecimal equivalent. 36

Example: Decimal → Hexadecimal Example: Convert the decimal number 42910 into its hexadecimal equivalent. 42910 = 1 AD 16 = 1 ADH 37

Hexadecimal ↔ Decimal Process The Process: Weighted Multiplication Multiply each bit of the hexadecimal number by its corresponding bit-weighting factor (i. e. , Bit-0→ 160=1; Bit-1→ 161=16; Bit-2→ 162=256; etc. . . Add up all of the products in step (a) to get the decimal number. 38

Hexadecimal ↔ Decimal Process Example: Convert the octal number 5 E 16 into its decimal equivalent. 5 E 161 160 16 1 80 + 14 = Bit-Weighting Factors 9410 39

Hexadecimal ↔ Decimal Process Example: Convert the octal number 5 E 16 into its decimal equivalent. 5 E 161 160 16 1 80 + 14 = Bit-Weighting Factors 9410 40

Hexadecimal ↔ Decimal Process Example: Convert the octal number 5 E 16 into its decimal equivalent. 5 E 161 160 16 1 80 + 14 = Bit-Weighting Factors 9410 41

Hexadecimal ↔ Decimal Process Example: Convert the octal number 5 E 16 into its decimal equivalent. 5 E 161 160 16 1 80 + 14 = Bit-Weighting Factors 9410 42

Hexadecimal ↔ Decimal Process Example: Convert the octal number 5 E 16 into its decimal equivalent. 5 E 161 160 16 1 80 + 14 = Bit-Weighting Factors 9410 43

Hexadecimal ↔ Decimal Process Example: Convert the octal number 5 E 16 into its decimal equivalent. 5 E 161 160 16 1 80 + 14 = Bit-Weighting Factors 9410 44

Hexadecimal ↔ Decimal Process Example: Convert the octal number 5 E 16 into its decimal equivalent. 5 E 161 160 16 1 80 + 14 = Bit-Weighting Factors 5 EH = 9410 45

Example: Hexidecimal → Decimal Example: Convert the hexadecimal number B 2 EH into its decimal equivalent. B 2 E 162 161 160 256 16 1 2816 + 32 + 14 = 286210 46

Example: Hexidecimal → Decimal Example: Convert the hexadecimal number B 2 EH into its decimal equivalent. B 2 E 162 161 160 256 16 1 2816 + 32 + 14 = 286210 47

Example: Hexidecimal → Decimal Example: Convert the hexadecimal number B 2 EH into its decimal equivalent. B 2 E 162 161 160 256 16 1 2816 + 32 + 14 = 286210 48

Example: Hexidecimal → Decimal Example: Convert the hexadecimal number B 2 EH into its decimal equivalent. B 2 E 162 161 160 256 16 1 2816 32 14 = 286210 49

Example: Hexidecimal → Decimal Example: Convert the hexadecimal number B 2 EH into its decimal equivalent. B 2 E 162 161 160 256 16 1 2816 + 32 + 14 = 286210 50

Example: Hexidecimal → Decimal Example: Convert the hexadecimal number B 2 EH into its decimal equivalent. B 2 E 162 161 160 256 16 1 2816 + 32 + 14 B 2 EH = 286210 51

Example: Hexidecimal → Octal Convert the hexadecimal number 5 AH into its octal equivalent. 52

Example: Hexidecimal → Octal Convert the hexadecimal number 5 AH into its octal equivalent. First convert the hexadecimal number to its decimal equivalent, then convert the decimal number to its octal equivalent. 53

Example: Hexidecimal → Octal Convert the hexadecimal number 5 AH into its octal equivalent. First convert the hexadecimal number to its decimal equivalent, then convert the decimal number to its octal equivalent. 5 A 161 160 16 1 80 + 10 = 9010 54

Example: Hexidecimal → Octal Convert the hexadecimal number 5 AH into its octal equivalent. First convert the hexadecimal number to its decimal equivalent, then convert the decimal number to its octal equivalent. 5 A 161 160 16 1 80 + 10 = 9010 5 AH = 1328 55

Example: Octal → Binary Convert the octal number 1328 into its binary equivalent. 56

Example: Octal → Binary Convert the octal number 1328 into its binary equivalent. First convert the octal number into its decimal equivalent, then convert the decimal number into its binary equivalent. 57

Example: Octal → Binary Convert the octal number 1328 into its binary equivalent. First convert the octal number into its decimal equivalent, then convert the decimal number into its binary equivalent. 1 3 2 82 81 80 64 8 1 64 + 2 = 9010 58

Example: Octal → Binary Convert the octal number 1328 into its binary equivalent. First convert the octal number into its decimal equivalent, then convert the decimal number into its binary equivalent. 1 3 2 82 81 80 64 8 1 64 + 2 = 9010 1328 = 10110102 59

Binary ↔ Octal ↔ Hex Shortcut Because binary, octal, and hex number systems are all powers of two (which is the reason we use them) there is a relationship that we can exploit to make conversion easier. 1 0 1 0 2 = 132 8 = 5 A H To convert directly between binary and octal, group the binary bits into sets of 3 (because 23 = 8). You may need to pad with leading zeros. 0 0 1 1 0 2= 1 3 28 1 3 2 001 010 To convert directly between binary and hexadecimal number systems, group the binary bits into sets of 4 (because 24 = 16). You may need to pad with leading zeros. 0 1 1 0 2= 5 5 A A 16 0101 1 010 60

Example: Binary ↔ Octal ↔ Hex Example: Using the shortcut technique, convert the hexadecimal number A 616 into its binary and octal equivalent. Use your calculator to check your answers. 61

Example: Binary ↔ Octal ↔ Hex Example: Using the shortcut technique, convert the hexadecimal number A 616 into its binary & octal equivalent. Use your calculator to check your answers. Solution: First convert the hexadecimal number into binary by expanding the hexadecimal digits into binary groups of (4). A 6 16 1010 0110 A 616 = 101001102 Convert the binary number into octal by grouping the binary bits into groups of (3). 010100110 2 4 6 101001102 = 2468 62